Economics Working Paper Series 2020/013 The acceptability of lotteries in allocation problems: a choice-based approach Elias Bouacida and Renaud Foucart The Department of Economics Lancaster University Management School Lancaster LA1 4YX UK © Authors All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission, provided that full acknowledgement is given. LUMS home page: http://www.lancaster.ac.uk/lums/ The acceptability of lotteries in allocation problems: a choice-based approach Elias Bouacida∗ Renaud Foucart† July 20, 2020 Abstract We report the results of two experiments on the social acceptability of random devices in allocation mechanisms. A majority of subjects do not opt for a lottery if they can ra- tionalize an alternative mechanism as non-random. It is, however, possible to design a payoff-equivalent mechanism to the lottery that is more acceptable. Our results shed light on the real-world reliance on obscure criteria in allocation problems where lotteries seem to be simpler and more efficient. Keywords: lotteries, mechanism design JEL-Code: D01, D78, D91 1 Introduction Consider an allocation problem in which a subject can either receive a high or a low reward. This problem can be solved either with a lottery or with a set of criteria. If the two methods offer the same odds of getting the high reward, will the subject prefer one over the other? In this paper, we report the results of two experiments revealing a preference for the use of criteria. The preference is not explained by overconfidence but seems to originate from an aversion to the explicit use of a lottery. Reframing the lottery by allowing subjects to take a payoff-irrelevant decision makes it more acceptable. Many of the solutions provided by economists to real-world allocation problems involve a role for randomization, usually as a tie-breaking rule. A leading example is the school allocation mechanisms, such as the deferred acceptance algorithm.1 In practice, however, allocations problems solved using explicit lotteries are rare (we discuss some of them in Appendix A). For instance, in USA law, the Administrative Procedure Act (APA) forbids an agency to take any action that is “arbitrary, capricious, or an abuse of discretion” (Vermeule, 2015, p.475). It means that even in the absence of any information or rational reason to pick one decision over the other, a federal agency is compelled to provide a rationalization and can never toss a coin.2 Anecdotal evidence from the implementation of school choice mechanisms in several countries ∗Lancaster University Management School, [email protected] †Lancaster University Management School, [email protected] 1Debates on the use of randomization in school choice revolve on its impact on parents’ strategies (Erdil and Ergin, 2008; Kesten and Ünver, 2015; Abdulkadiroglu et al., 2019), but not on the acceptability of randomization in itself. 2What lawmakers find objectionable is not the fact that an agency has the same probability of reaching one decision or the other, but the failure to deliberate that the use of a randomization device represents (Leib and Galoob, 2016, p.1860). In a case of custody allocation, the supreme court of Virginia stated that “A judge’s act of tossing a coin in a courtroom to decide a legal issue pending before the court suggests that courts do not decide cases on their merits but instead subject litigants to games of chance in serious matters without regard to the evidence or applicable law.” (Supreme Court of Virginia, Record No. 071014). 1 also shows a reluctance of parents to accept lotteries and a preference for “objective” albeit often arbitrary and murky criteria.3 Another example is the award of research funding by peer review. Except for the very best and worst submissions, this procedure is largely unpredictable (Cole et al., 1981; Graves et al., 2011; Pier et al., 2018). Given the high cost spent writing and evaluating proposals, it would be more cost-efficient to allocate funding randomly among applications deemed good enough, perhaps with a subset of outstanding proposals guaranteed funding (Greenberg, 1998; Brezis, 2007; Fang and Casadevall, 2016; Roumbanis, 2019; Avin, 2019). However, in practice, scientists and funding bodies have resisted this approach, insisting on providing a rationalization (Barnett, 2016).4 We ran two incentivized experiments on Amazon Mechanical Turk (AMT hereafter) to study whether there exists an individual revealed preference for the use of criteria over lotteries. Subjects were paid a show-up fee to participate and earned an additional reward based on the result of a game. A necessary step to be able to identify a potential preference was to identify a set of criteria for which we could provide exact probabilities of winning before subjects chose between it and a lottery. In both experiments, we asked subjects to give a sequence of 5 strategies of rock-paper-scissors (RPS) at the beginning. The choice of RPS strategies could not be modified further. It became a set of criteria for which the odds of winning against a given set of strategies could be computed. We then asked subjects to choose the process that determined if they won the reward. The first possibility was to use their chosen strategies against a player drawn from a data set of 2,500 real-world RPS players. They won the reward if they won more RPS games than their randomly drawn opponent. In case of ties, the first who won a round won the game.5 In experiment 1, the second possibility was to condition the reward on the toss of a fair coin by an algorithm (CT hereafter). We chose this device as the simplest and most familiar lottery, thereby more likely to be perceived as fair (Eliaz and Rubinstein, 2014). We find that, on average, 66% of the subjects chose the RPS over the CT. When we provided subjects their odds of winning RPS given their chosen strategies, some reacted to this information. On average, our regression analysis estimates that out of 100 subjects, 25 switch mechanism depending on whether or not their RPS strategies have a probability of winning higher than 1/2. Subjects knowing their odds of winning RPS were on average as likely to choose the coin toss that those who did not know it, showing that the preference for RPS is not explained by overconfidence. Moreover, subjects reacted similarly to the actual odds of winning and to their perceived probability of winning when deciding under uncertainty. In experiment 2, on top of 5 RPS strategies, we asked subjects to provide 5 strategies of a repeated matching pennies (MP) game. The MP strategies would be played against a computer choosing head and tails with equal probability. We then offered subjects the same choice as in 3In France, the system of allocation of students to universities (APB) used lotteries to break ties when capacity was reached. It was criticized and is now replaced with another mechanism with a lot more criteria, reducing the likelihood of ties and ending the use of lotteries. Importantly, the weights between the criterion are not public and leave a lot of almost arbitrary leeway to the universities, which was not the case previously. In French-speaking Belgium, a 2009 attempt to randomly allocate students in oversubscribed high schools lasted only a year after being dubbed by some parents and the media the “lottery law” and replaced by a set of criteria still in use today. In the UK, when a lottery was implemented for school allocation in Brighton and Hove, the Conservative party (then in the opposition) pledged a country-wide ban of lotteries. On March 2008, before becoming Secretary of State for Education, Michael Gove justified this stance on the BBC as “Would you like the fate of your children to be decided by the spin of the roulette wheel or the roll of the dice? No, and neither would I. I think that one of the key problems with lotteries is that they reduce almost one of the most important decisions that any parent is going to make for their child, into a matter of chance.” While the ban was never implemented, lotteries have not been extended either. 4The Health Research Council of New Zealand is a notable exception, running a pilot random allocation for its early career “Explorer” grants since 2013. 5And if there was still a tie, in the end, the player won if she had spent an even number of seconds choosing the RPS strategies. 2 experiment 1 but using MP instead of CT as an alternative to RPS. MP and CT are in practice equivalent. MP’s odds are not influenced by a subject’s choice of strategies. We find, however, that only 42% of our subjects chose the RPS in this setting, a proportion significantly different from experiment 1. Re-framing the lottery as the result of a choice, even when the expected payoff is the same, seems to make it much more socially acceptable. This result also suggests that the preference for RPS in experiment 1 is not based on that game being factually more “meritocratic”, as the probability of winning the MP is independent of the chosen strategies. It also rules out the “competence hypothesis” of Heath and Tversky (1991) as an explanation. There is no reason subjects would feel relatively more competent in experiment 2 with MP than they are with CT in experiment 1. The novelty of our approach is to provide an incentivized choice between a lottery and a set of criteria. Survey evidence shows that people are reluctant to rely on the use of a random device to determine the outcome of important hypothetical decisions involving other people (Keren and Teigen, 2010).